The Labor Market Effects of Reducing the Number of Illegal Immigrants

Working Paper 04-2015 The Labor Market Effects of Reducing the Number of Illegal Immigrants Andri Chassamboulli and Giovanni Peri Department of Economics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus Tel.: +357-22893700, Fax: +357-22895028, Web site: http://www.ucy.ac.cy/econ/en

The Labor Market Effects of Reducing the Number of Illegal Immigrants Andri Chassamboulli (University of Cyprus) Giovanni Peri (University of California, Davis)∗ March, 23rd, 2015 Abstract A controversial issue in the US is how to reduce the number of illegal immigrants and what effect this would have on the US economy.

To answer this question we set up a two-country model with search in labor markets and featuring legal and illegal immigrants among the low skilled. We calibrate it to the US and Mexican economies during the period 2000-2010. As immigrants, especially illegal ones, have a worse outside option than natives their wages are lower. Hence their presence reduces the labor cost of employers who, as a consequence, create more jobs per unemployed when there are more immigrants. Because of such effect our model shows that increasing deportation rates and tightening border control weakens the low-skilled labor markets, increasing unemployment of native low skilled.

Legalization, instead decreases the unemployment rate of low-skilled natives and it increases income per native.

JEL codes: F22, J61, J64. Key Words: job creation, search costs, illegal immigrants, border controls, deportations, legalization, unemployment, wages. ∗ Andri Chassamboulli, Departement of Economics, University of Cyprus, CY-1678 Nicosia; andricha@ucy.ac.cy. Giovanni Peri, Department of Economics, UC Davis. One Shields Avenue, Davis Ca 95616 USA; gperi@ucdavis.edu. We are grateful to three anonymous referees for their helpful comments. 1

1 Introduction Most of the existing papers on the labor market effects of immigration consider the number and the skill composition of immigrants as an exogenous variable and analyze the consequences of changing those on native labor market outcomes.

The number and type of immigrants entering a country, however, are not policy variables of choice, but the outcomes of economic, social and policy forces in the sending and receiving countries. In the economic literature on the effect of immigration, very little attention has been paid to the specific policies used and to the difference between the labor market effects of legal and illegal immigrants.1 Large part of the policy debate in the US, however, has been about different ways to reduce the number of illegal immigrants. The presence of a large number of illegal immigrants is an anomaly, but there is disagreement on how to address it.

A question often asked to economists is whether reducing illegal immigrants would be costly or beneficial for the US economy. In particular, what policy, among border enforcement, deportation, self-deportation or legalization, would be most harmful to US firms and workers? The existing economic analysis uses naive frameworks to answer this question. Based on an oversimplified canonical model of labor demand and supply, economists rarely focus explicitly on illegal immigrants and they overlook the different implications across policies. The goal of this paper is to fill this gap by using a more insightful model to analyze the effects of different policies aimed at reducing the number of illegal immigrants.

Do fewer illegal immigrants free jobs for Americans or do they reduce firm’s profit and job creation? Will legalization increase migration pressures? Will deportation and border control decrease legal immigration?

To address these important questions we propose a new model representing two connected labor markets, parameterized to match US and Mexico, and two groups of workers, high-skilled and low-skilled, that are complementary in production. Firms create jobs that are skill-specific and search frictions in the market exist. Legal and illegal migration opportunities from Mexico to the US arise and people take them if they increase their expected labor income net of costs. To focus on the issue of illegal immigrants we consider migration of low-skilled workers only from Mexico, while US workers can be low-skilled 1 Throughout the paper we will use the adjectives “legal” and “illegal” immigrants to characterize immigrants who are endowed or not of proper documentation to reside and work in the US.

Some scholars refer to those groups as “regular” and “irregular” or as “documented” and “undocumented” immigrants. 1

(competing with immigrants) or high-skilled (complementing them in production). This model incorporates aspects of labor markets and migration, which would not be captured by a classical demand-supply framework and turn out to be crucial. First, we characterize legal and illegal immigrants and native workers in the receiving country (US) as potentially different in their outside options and in their probability of breaking up a job-match. These differences affect the wage that each type of worker can bargain with firms, for given productivity of the worker-firm match. In particular, illegal immigrants usually have the worse outside option, followed by legal immigrants and then natives.

Hence the first group will accept lower wages, relative to legal immigrants and natives, which imply that US firms can cut labor costs by hiring them. Second, as a consequence of these labor cost savings, firms are willing to post more job openings and if those are specific to skills, but not to immigrant workers, a positive job-creation effect will benefit native employment opportunities too.

Given the large productivity difference between Mexico and the US, illegal immigration opportunities, albeit associated to worse conditions than legal ones, can be attractive to Mexican unskilled workers. At the same time US firms benefit from illegal immigrants by paying lower labor cost. These features capture the economic incentives that have lead to illegal immigration in the US. However, there is also another crucial implication of this framework: besides rich country skilled workers, also unskilled workers, can benefit from illegal immigrants. More illegal workers push firms to create more jobs per unemployed worker in the unskilled labor markets because their presence reduces the average firm’s cost.

As long as labor markets are not fully segmented between immigrants and natives, natives will increase their employment too. Hence policies reducing the number of illegal immigrants may cost employment and income to natives. With our model we can quantify these costs and analyze how policies differ from each other. The four policies that we analyze are the following: (i) increasing border enforcement to reduce illegal immigration opportunities (ii) increasing the costs that illegal immigrants face in looking for a job (no access to benefits), (iii) increasing the frequency of deportations and (iv) increasing the probability of legalization.

In analyzing these policies we take a status-quo driven approach. Rather than asking whether there is a theoretically optimal number of illegal immigrants from the perspective of native income per person, we consider the status quo, and we ask for each policy what would be the cost, in terms of native income per person, wage and employment of reducing the number of illegal 2

immigrants by a certain percentage. The policies described can be separated into two categories. Three of them (increased deportation, increased border control, and increased cost of looking for a job) not only decrease the number of illegal immigrants, but they also reduce the total number of immigrants (legal plus illegal). We will call these three “restrictive policies”. To the contrary legalization, the fourth policy, decreases the number of illegal immigrants, but it increases the number of total immigrants. By turning illegal immigrants into legal, this policy leaves the total immigrant stock unchanged and it also provides stronger incentives for potential immigrants, as more of them can become legal in the US.

The three restrictive policies, by reducing unskilled immigrants (illegal and total) have a depressing effect on wage and employment of skilled workers (complementary to unskilled), and on firms’ profits (that benefit from the cost-reducing effects of illegal immigrants). In the canonical model, however, they would increase employment and wages of native unskilled, by reducing competition from unskilled immigrants. To the contrary, because of the unskilled-job creation effect of immigrants described above, the restrictive policies worsen labor market conditions for unskilled natives when analyzed within our model.

Legalization, instead, as it increases the total number of unskilled immigrants enhancing their job-creation effect, produces a positive effect on wages and employment of skilled natives and a positive effect also on unskilled native employment. While the wage effect of legalization on unskilled natives is negative, the overall effect on income per native in the receiving country is positive, contrarily to the restrictive policies that reduce income per native in the receiving economy.

Quantitatively our simulations shows the following effects. Increasing the deportation rate of illegal immigrants or reducing illegal immigration opportunities at the border, to achieve a 50% reduction in the number of illegal immigrants (a very aggressive program) would produce an increase in the unemployment rate of unskilled natives by about 1.13% of its initial value. The unemployment rate of native skilled workers also increases by 0.57%. The first effect is due to a decrease in unskilled job creation by firms and the second to the negative productivity effect on skilled workers due to complementarity.

A similar result obtained through increasing the cost of unemployment for illegal immigrants, would generate qualitatively similar but somewhat smaller results on unskilled native unemployment (+0.95%). This is because that policy would also reduce the wage of illegal immigrants and hence partly offset the negative incentives in creating unskilled 3

jobs. The same reduction in illegal immigrants achieved with a legalization program would produce very different effects. The unemployment rate of unskilled natives would decrease by about 1.31% of its initial value and that of skilled native would decrease by 1.20% of its initial value. The increase in legal immigrants generated by the legalization program turns the negative labor market effect into a positive one. At the same time legalization is the only policy that increases income per native in the scenario presented above (+0.45%), while the losses of high skilled native workers, of firm profit and the employment loss of low-skilled natives result in a net decrease in income per native when adopting the other three policies (−0.25/ − 0.28%).

Several checks on parameter values and on different scenarios about immigrants’ productivity confirm that the above results are quite stable and they apply, qualitatively, to most plausible scenarios. In summary, while the effects on income and unemployment are quite small, the difference between the restrictive policies (that deliver similar effects within each other) and the legalization is very clear: legalization is the only policy that produces an increase in income per native and a decrease in native skilled and unskilled unemployment. As the administrative costs to implement legalization are also likely much smaller than those of increasing border security and certainly of those of deporting immigrants our analysis suggests that in terms of consequences on income of natives legalization seems the best option.

This paper is related to a large empirical literature on the effect of immigration on US labor market outcomes (see the meta-analysis by Longhi, Nyikamp and Poot (2005), (2008) and Lewis and Peri (forthcoming) for reviews of several important recent findings). Most of that literature adopts a canonical neoclassical labor demand-supply approach to derive a reduced form equation (e.g. Borjas 2003) or a slightly more structural approach to estimate the elasticity of relative demand (Ottaviano and Peri 2012, Manacorda et al 2012). Very few studies analyze immigration within the context of search-matching models of the labor market.

Even fewer differentiate between legal and illegal immigration when looking at labor market implications.

The paper most closely related to ours is Chassamboulli and Palivos (2014). In that paper, however, immigration is exogenous, only the receiving country is analyzed, only legal immigrants exists and no policy is explicitly considered. Chassamboulli and Palivos (2014) is the first paper, to our knowledge, that introduces the important job-creation effect of immigrants stemming from the fact that the profit for the firm generated by 4

immigrants is larger than that generated by natives. This is an important building block of our model too. We add to that framework the very important difference between legal and illegal immigrants, the modelling of migration decision from Mexico, and the representation and analysis of specific policies.

With those tools we are the first to analyze the income and employment impact of different policies reducing the number of illegal immigrants. Palivos (2009) is one of the very few papers analyzing the welfare effects of illegal immigrants on natives. Liu (2010) is the only other model we are aware of, that analyzes the effects of illegal immigration on the receiving country using a search and matching model. In his model Liu (2010) only includes illegal immigrants and assumes that they are identical to natives in their search and labor supply behavior, but may be complementary to native workers in production.

We consider, instead that immigrants, particularly illegal ones, are disadvantaged relative to natives in terms of job search conditions and search costs (they receive lower or no benefits when unemployed) and we also include the possibility that illegal immigrants are subject to the risk of deportation. In our model what is commonly referred to as “exploitation” of illegal immigrants, namely them being paid lower salaries, is due to their worse bargaining position vis-a-vis their employer relative to natives.

Finally, somewhat related to this paper, although mainly empirical, is the literature on immigration and labor market institutions. It has been recognized for some time that the specific labor market institutions (level of unemployment benefits, costs of hiring, centralization of wage bargaining) can affect significantly the impact of immigration on employment and wages of natives. For instance Angrist and Kugler (2003) show that more protective labor markets result in larger impact of immigration on unemployment. D’Amuri and Peri (2014) also show that labor reallocation and the complementarity effects of immigrants can be larger in markets with lower rigidities.

The rest of the paper is organized as follows. Section 2 presents the model and provides intuition for its main results and the working of different mechanisms. We then describe in Section 3 the policy experiments that we will be considering and two special cases that allow us to illustrate the functioning of two important mechanisms in the model. Section 4 describes the parameterization of the model calibrated to match the main labor market statistics of the US and Mexico for the period between 2000 and 2010. Section 5 shows the main effects obtained by simulating four different policies that would achieve a 5

reduction of the number of illegal immigrants in the rich country. In Section 6 we present some checks that the results are robust to reasonable variations of the parameter values. Section 7 concludes the paper. 2 The Model We describe here the main features of the model. Details of the equilibrium conditions and derivation of intermediate results are described in the Appendix (A). We consider two countries indexed by i = [1, 2]. Each country is endowed with a continuum of workers. All agents are risk neutral and discount the future at a common rate r > 0, equal to the real interest rate, time is continuous.

Country 1 has higher wages and higher employment rate than country 2. Hence workers have economic incentives to migrate from country 2 to country 1. No worker has incentives to migrate from country 1 to country 2. Migration can be legal or illegal. We denote with I and L, respectively, the number of illegal and legal migrant workers in country 1. The difference between the two is that opportunities to migrate illegally are more frequent than those to migrate legally. However illegal immigrants have higher search costs in the labor market and they face risk of deportation. The size of the labor force native of country 1 (indicated as N) is normalized to 1 and it is divided into two types of workers: skilled in measure of S and unskilled in measure of 1−S.

Individuals born in country 2 are, instead, of measure F (foreign) and we assume that they are all unskilled. The reason for this simplification is that we are focussing on the MexicoUS migration which mainly involves unskilled workers (without tertiary education). The total labor force of country 1 consists of natives, legal and illegal immigrants and its size is 1 + I + L. The size of total labor force in country 2 is F − I − L. Individuals from either country enter the labor force at rate τ and they exit at rate τ, so that the overall size of the labor force (native of country 1 and 2) remains constant.

The new individuals enter the labor force as unemployed.

At any point in time, opportunities to migrate arise as “random events” occurring at rate µx, if the worker is unemployed in country 2, and at rate µe x, if the worker is employed. The subscript x = [I, L] indicates the type of the immigration opportunity. Specifically, the worker may find an opportunity to migrate to country 1 legally (L) or illegally (I). Once in country 1, illegal immigrants face some risk of deportation but they may obtain legal status with probability n. This reflects the possibility that through some special circumstances (e.g. marriage) some illegal immigrants may become legal.

  • This probability is however very small in absence of a legalization program. We assume that µx > µe x and without loss of generality we choose µe x = 0, x = [L, I]. Migration opportunities, that is, arise only for the unemployed, who are actively looking for them. This captures the idea that, in order to migrate, workers often need to move closer to the border and actively look for migration opportunities. A worker will act upon an opportunity to migrate to country 1 if the benefit exceeds the cost. The migration cost, z is heterogeneous across individuals and it is distributed according to the CDF Φ(z) with support [z, z̄]. Only the fraction of workers with costs lower than expected benefits will migrate. Once in country 1, migrants search for a job. Hence, the benefit from immigrating to country 1 is the difference between the value of searching for a job as an immigrant in country 1 and the value of searching for a job as a native in country 2. 2.1 Workers and Firms Firms in country 1 operate in one of two intermediate sectors or in the final sector.2 The two intermediate sectors produce intermediate goods Y u 1 and Y s 1 using “unskilled” and “skilled” labor, respectively. Each of these two sectors operates a linear technology, which, through normalization of units, yields output equal to the number of the respective workers employed. These intermediate inputs are non-storable. Once produced, they are sold in competitive markets and are assembled for the production of country’s 1 final good (Y1) which is also the numeraire. The production technology for the final good of country 1 is as follows: Y1 = [α(Y s 1 )ρ + (1 − α)(Y u 1 )ρ ]1/ρ , ρ ≤ 1, (1) where α is a positive parameter that governs income shares and ρ determines the elasticity of substitution between the unskilled and skilled inputs. Since the two intermediate inputs are sold in competitive markets, their prices, ps 1 and pu 1 will be equal to their marginal products, that is: ps 1 = α
  • Y1 Y s 1
  • 1−ρ , (2) pu 1 = (1 − α)
  • Y1 Y u 1
  • 1−ρ , (3) 2 Our production side borrows from Acemoglu (2001). 7

The production technology in (1) implies diminishing marginal products and Edgeworth complementarity between the two inputs Y s 1 and Y u 1 .3 The migrants from country 2 in country 1 supply labor to the unskilled intermediate sector. The natives, on the other hand, can be either skilled (s) or unskilled (u). Hence the skilled labor market in Country 1 hires only skilled native workers whose marginal productivity is ps 1 and the unskilled labor market hires unskilled native workers and immigrants with marginal productivity pu 1. The production technology in (1) implies that immigrants are complements for skilled native workers and perfect substitutes for unskilled native workers.

Without loss of generality, we keep the economy of country 2 simple by assuming that all workers in country 2 are identically unskilled. There is therefore only one labor market in country 2 in which all matches produce a constant output p2 and total output in that country is equal to Y2 = (F U2)p2, where U2 denotes the unemployed labor force of country 2 and is defined below.

2.2 Search and Matching In each labor market of country i unemployed workers and unfilled vacancies are brought together via a stochastic matching technology Mi(Ut ), where t = [u, s] denotes the skill-type. Ut i and V t i denote, respectively, the number of unemployed workers and vacancies of skill t in country i.4 We assume that the function Mi(Ut ), i = [1, 2] exhibits standard properties: it is at least twice continuously differentiable, increasing in its arguments, it exhibits constant returns to scale and satisfies the Inada conditions. Using the property of constant returns to scale, we can write the flow rate of match per unemployed worker of skill type t in country 1 as Mi(Ut )/Ut i = mi(θt i).

The flow rate of match per vacancy is Mi(Ut )/V t i = qi(θt i), where θt /Ut i = mi(θt i)/qi(θt i) represents the measure of tightness in market t of country i and mi(θt i) is increasing in θt i while qi(θt i) is decreasing in θt i.

Each firm posts at most one vacancy. The number of vacancies in each market is determined endogenously by free entry. While vacancies in country 1 are skill-specific, they cannot be specifically “targeted” to natives or to immigrants. They are open to both native and immigrant workers with those skills. A vacant firm bears a recruitment cost ct i specific to the country and skill type, related to the expenses of keeping a vacancy 3 That is: ∂pt 1 ∂Y t 1 < 0 and ∂px 1 ∂Y t 1 > 0 for x 6= t. 4 Since there is only one labor market in country 2 the superscript t is not relevant in the case i = 2.

In what follows we therefore drop the superscript t whenever i = 2.

open and looking for a worker. An unemployed worker of type t in country i receives a flow of income bt i, which can be considered as the opportunity cost of employment. In addition, unemployed workers pay a search cost πt ij per unit of time where the subscript j = [N, I, L] denotes the worker’s origin and status: native (N), illegal immigrant (I) and legal immigrant (L). Such subscript applies only to the unskilled market of country 1. We account for the fact that a legal immigrant worker faces a higher search cost compared to a native workers and an illegal immigrant faces even higher costs.

The reason is that legal immigrants, whether on temporary visas or permanent resident have access to significantly fewer benefits than US citizens. Since the Personal Responsibility and Work Opportunity Reconciliation Act (PRWORA) of 1996 many federal government benefits (Food stamps, TANF, AFDC and others) were restricted to US citizens only. Hence non-naturalized legal immigrants (the majority of unskilled foreign-born) had a significant larger cost of being without a job. In the 2000’s some but not all, states re-instated some of them. Moreover all legal immigrants on temporary visas (such as H2B and other working visas) are not eligible for any welfare assistance, including unemployment insurance.

Hence their access to income when not employed is significantly smaller than for natives. Undocumented immigrants cannot access any welfare program/unemployment insurance at all and hence their cost of searching is even larger. We standardize the search cost of a native worker to 0 and we set πs 1N = πu 1N = π2N = 0, πu 1I = πI, πu 1L = πL and we presume πI > πL > 0 which will be confirmed by the calibration. Legal immigrants face zero deportation risk. They have a positive probability of returning home, however, reflecting the possibility of return for personal or other reasons. Illegal immigrants face the additional risk of being repatriated by deportation.

Hence the return probability of illegal immigrants is higher than that of legal immigrants. Let dL and dI denote the instant return rate of legal and illegal immigrants, respectively. We set dI ≥ dL > 0 where their difference is the deportation rate. Upon return to country 2 the worker joins the pool of unemployed and starts searching for a job. When a vacancy and a worker are matched, they bargain over the division of the produced surplus. The status of the worker as well as the output that results from a match are known to both parties. Wages, denoted as wt ij, differ by country (i), skill type (t) and migration status (j).

They are determined by Nash bargaining of the produced surplus between the firm and the worker. After an agreement has been reached, production commences immediately. Matches in country i dissolve at the rate σt i, specific to skill 9

type t and country i. Following a job destruction, the worker and the vacancy enter the corresponding market and search for new match. 2.3 Optimality Conditions and Free entry At each point in time a worker is either employed (E) or unemployed (U), while a vacancy may be either filled (F) or empty (V ). We use the notation Jκ,t ij to denote the present discounted value associated with each state κ = [V, F, U, E], where i = [1, 2] denotes the country, j = [N, I, L] the worker’s immigration status and t = [u, s] indicates the worker’s skill type.

Eighteen Bellman equations describe the optimal behavior of workers and firms.

Since all workers and firms in country 2 are identical, four Bellman equations (one for each state κ = [V, F, U, E]) describe the values of workers and firms in country 2. The remaining fourteen Bellman equations describe the values of workers and firms in country 1, where workers differ in terms of skills and immigration status. Specifically, for each of the three states [F, U, E] there are four Bellman equations: one for legal immigrants, one for illegal immigrants one for unskilled natives and one for skilled natives. The value of an unskilled vacancy searching for a worker (V ), instead, is the same for legal immigrants, illegal immigrants and unskilled natives because the vacancy is open to any of them and hence it is described by the same Bellman equation.

Another Bellman equation describes the value of a skilled vacancy.5 The full set of Bellman equations is in the Appendix A. A second set of equilibrium conditions is that of free-entry (vacancy posting) on the firm side in each of the two labor markets in country 1 (skilled and unskilled) and in country 2. Firms open vacancies up to the point that an additional one has zero expected value. In equilibrium this implies the following three conditions: JV,t i = 0, i = [1, 2] and t = [s, u] if i = 1, (4) Wages are then determined by Nash bargain between the matched firm and the worker. The outside options of the firm and the worker are the value of a vacancy and the value of being unemployed, respectively.

Let St ij ≡ JF,t ij + JE,t ij − (JU,t ij + JV,t i ) denote the surplus of a match between a vacancy of skill type t in country i and a worker of immigration 5 The superscript t and the subscript j are not relevant for country 2, we therefore drop them whenever i = 2. We also drop the superscript t in the cases j = [L, I], since all immigrants provide only unskilled labor and can only be employed in unskilled jobs, and the subscript j in the case κ = V and i = 1, since unskilled vacancies in country 1 are common to immigrants and natives.

status j. With Nash-bargaining the wage wt ij is set to a level such that the worker gets a share β of the surplus, where β represents the relative bargaining power of workers, and the share (1 − β) goes to the firm. This implies five equilibrium conditions (for matches with legal immigrants, illegal immigrants, unskilled natives and skilled natives in country 1 and for matches with native workers in country 2) of the following form: βSt ij = JE,t ij − JU,t ij (1 − β)St ij = JF,t ij − JV,t i (5) for i = [1, 2]; j = [N, I, L] if i = 1; and t = [s, u] if j = N 2.4 The Immigration Decision An (unemployed) worker located in country 2 will choose to immigrate to country 1, when an immigration opportunity arises, if its benefit exceeds its cost.

The benefit from migration is the difference between the value of searching for an unskilled job in country 1 and the value of searching in country 2. Workers are heterogeneous in their migration costs. A worker whose migration cost is z, will chose to take advantage of an opportunity to enter legally in country 1 only if JU 1L − JU 2 ≥ z while he/she will enter illegally if JU 1I − JU 2 ≥ z. The threshold costs, denoted as z∗ I and z∗ L, and representing the highest cost a worker is willing to pay in order to obtain illegal or legal entry into country 1, are defined by the following conditions: z∗ I = JU 1I − JU 2 (6) z∗ L = JU 1L − JU 2 (7) Notice that in equilibrium z∗ L > z∗ I because the value of searching for a job in country 1 is higher when the immigrant is legal than when he/she is illegal (i.e.

JU 1L > JU 1I).

This proceeds from the assumptions that illegal immigrants have higher search costs (πI > πL > 0) and face the risk of deportation (dI > dL > 0) both of which reduce the value they can generate while searching for a job and the value of a job for them. This implies that for a given distribution of the migration cost z, there will always be a larger share of the country 2 population willing to take a legal immigration opportunity than an illegal one. 2.5 The Steady-State conditions The last set of equilibrium conditions are the steady-state conditions. Five of them determine the constant number of unemployed workers of each type in each country by 11

equating the flows into and out of unemployment status for each type of worker: U2 are in country 2, Us 1N are skilled natives in country 1, Uu 1N are unskilled natives in country 1, U1L are legal immigrants in country 1 and U1I are illegal immigrants in country 1. Two more conditions guarantee the stationarity of the number of legal and illegal immigrants, L and I by equating the flows into and out of the group. The seven formal conditions defining these steady state variables are given by (38-44) in the Appendix A.2. Let us also define the variables φ ≡ Uu 1N /(Uu 1N + U1I + U1L) to be the share of native workers in the pool of unemployed unskilled workers of country 1 and λ ≡ U1L/(U1L +U1I) to be the share of legal immigrants among unemployed immigrants in country 1.

In equilibrium φ and λ are also constant. Writing the steady state conditions for unemployed and migrants as a function of parameters, labor market tightness in the respective markets (θs 1, θu 1 , θ2) and threshold costs z∗ I and z∗ L we obtain the following expressions: us 1N = Us 1N S = σs 1 + τ σs 1 + τ + m(θs 1) (8) uu 1N = Uu 1N 1 − S = σu 1 + τ σu 1 + τ + m(θu 1 ) (9) u1I = U1I I = σu 1 + τ + dI + n σu 1 + τ + dI + n + m(θu 1 ) (10) u1L = U1L L = σu 1 + τ + dL − n I L (1 − u1I) σu 1 + τ + dL + m(θu 1 ) (11) u2 = U2 σ2 + τ σ2 + τ + m(θ2) (12) L = µLΦ(z∗ L)u2(F − I) + nI dL + τ + µLΦ(z∗ L)u2 (13) I = µIΦ(z∗ I )u2(F − L) dI µIΦ(z∗ I )u2 (14) Expressions (8)-(14) reveal some important mechanisms at work in our model.

First, (13) and (14) show that the equilibrium number of migrants I and L depend negatively on the return probabilities (dI and dL), positively on the rates of migration opportunities (µI, µL), and positively on the threshold migration costs z∗ I and z∗ L. The latter implies that any economic and policy factor that increases the value of searching for a job in country 1 relative to country 2, encourages immigration and translates in larger stocks of legal L and illegal I immigrants in country 1. Second, the legalization rate (n) increases the steady state number of legal immigrants L and decreases the steady state number of 12

illegal immigrants I. Third, as customary in these models, unemployment rates increase with the relative separation probability σt i and decrease with the matching probability m(θt i) in the corresponding market.6 The impact of immigration policies on θt i, and in turn, on the matching probability m(θt i), is the main channel through which they can influence the unemployment rate of the native workers that participate in that market. Let us notice that once the constant equilibrium values of L, I, Us 1N , Uu 1N , U1L, U1I are determined then, a linear technology determines production of intermediates for country 1 so that: Uu 1N − U1L − U1I and Us 1N .

2.6 Equilibrium The eighteen Bellman equations (20-37), five Nash-Bargaining conditions (5), three free entry conditions (4), seven steady-state conditions (8-14) and two immigration-threshold conditions (6-7) plus 2 marginal productivity conditions (2, 3), the two linear production functions of intermediates and the aggregate production function of country 1 (1) and country 2 constitute the fourty-one equilibrium conditions determining the fourty-one endogenous variables of the model. These endogenous variables are the eighteen values of Jκ,t ij across countries, skills and immigration status, five wages (ws 1N , wu 1N , w1L, w1I, w2), three labor market tightness values (θu 1 , θs 1 , θ2) the number of unemployed and migrants of each type (I, L, Us 1N , Uu 1N , U1L, U1I, U2) the immigration cost thresholds (z∗ I , z∗ L) the marginal productivity of skilled and unskilled workers (pu 1, ps 1), the output of skilled and unskilled firms (Y and the final output of country 1 and 2 (Y1, Y2).

In the appendix A.3 we show how to derive some intermediate results and provide a description for how to solve the model in blocks. Given the fact, however, that some of the expressions are cumbersome and not very intuitive we omit those from the text. We will explain, instead, before calibrating and simulating the full model, the intuition behind two key mechanisms with the help of two special cases described in Section 3 below. 2.7 Three key conditions Before moving to the special cases, it is useful, to show three equilibrium relations that provide some intuition for the role of legal and illegal immigrants on unskilled job creation (vacancy posting) by firms in country 1.

6 The unemployment rates of illegal and legal immigrants, u1I and u1L, increase also with the probability of return dI and dL (respectively) and with the exit/entry rate τ. All those parameters, in steady state, act as separation rates. 13

  • Manipulating the Bellman equations of the value (to the firm) of a filled unskilled vacancy, JF,u 1N , JF 1L and JF 1I we can write the difference in value between a native-filled vacancy and a legal immigrant-filled one and between one filled by a legal and an illegal immigrant as follows: JF,u 1N − JF 1L = [w1L − wu 1N ] + dLJF,u 1N r + τ + σu 1 + dL (15) JF 1L − JF 1I = [w1I − w1L] + [dI − dL] JF 1L r + τ + σu 1 + dI + n (16) Expression (15) reveals that if w1L < w1N , which would be the case when legal immigrants have higher search cost than natives (worse outside option) then JF,u 1N < JF 1L as long as dL is small. So the value of a legal immigrant is higher than that of a native to the firm as long as, given their equal productivity, the wage paid to the immigrant is low enough relative to the native wage, to compensate for the larger probability that the immigrants ends the match because of return to the country of origin. Likewise condition (16) reveals that if w1I < w1L, because illegal immigrants have worse outside options than legal ones, then JF 1L < JF 1I as long as the difference between the return probabilities (dI − dL) which represent the deportation rate, is sufficiently small. Hence, low deportation rates and high search cost for illegal immigrants make them particularly valuable to the firm. And low return rates and high search cost for legal immigrants make them valuable to the firm. A negative value of expressions (15), (16) implies that legal and illegal immigrants may stimulate job creation. This vacancy creation effect can be seen by manipulating the free entry condition for unskilled vacancies in country one to get: cu 1 q(θu 1 ) = φJF,u 1N + (1 − φ)
  • λJF 1L + (1 − λ)JF 1I
  • (17) In this expression a larger share of immigrants among the unemployed (smaller value of φ) and a larger share of illegal ones among them (smaller value of λ) increase the value of the right-hand side, as long as (15) and (16) are negative, by shifting weight on JF 1I relative to JF,u 1N . This would imply more vacancy posting (free entry) and an increase in market tightness θu 1 to increase the left-hand side and reduce the right-hand side and maintain the equality (recall that q(θu 1 ) is decreasing in θu 1 ). This implies that a policy that decreases the share of both illegal and total immigrants in the labor force certainly depresses the labor market tightness through this channel. However a policy that decreases the share of 14

illegal immigrants but increases the share of total immigrants may offset the first negative impact with a positive impact on θu 1 . Finally let us notice that the impact of immigrants on θu 1 is also the channel through which they affect skilled native workers. For as long as skilled and unskilled workers are complementary in production, larger supply of the unskilled labor input Y u 1 , implies larger price for the skilled labor input ps 1 and thus larger profits for skilled firms. Hence immigration policies that stimulate the creation of unskilled jobs and raise θu 1 will also stimulate the creation of skilled jobs (i.e.

raise θs 1), with a positive impact on skilled native employment and wages.

3 Policy Effects in Special Cases The rich structure of the model presented above allows us to analyze different policies. We consider four of them: (i) reduced opportunities of illegal entry (increased border control) captured by a decline in µI; (ii) increased search cost for illegal immigrants, captured by an increase in πI; (iii) increased probability of deportation, captured by an increase of dI for given dL (iv) increased probability of legalization, captured by an increase in n. All these measures reduce the number of illegal immigrants. They have, however, different implications on native labor markets as well as different incentive effects on immigration.

There are the two main channels through which the presence of illegal (and legal) immigrants affects labor market outcomes of natives in our model. The first channel, that we call “price channel”, operates through the price of the intermediate input, pu 1.

As evidenced in equation (3) a decrease in I which is translated by the linear production technology into a decrease in Y u 1 increases the marginal productivity of the unskilled labor input thereby causing its price to rise. This “price effect” is the standard one, also present in the canonical model: immigrants are substitute for native unskilled and reducing their supply the marginal productivity of those increases putting upward pressure on their wages and downward pressure on their unemployment rate. The second channel, that we call “labor-cost channel” works instead through the expected labor cost to an unskilledsector firm from a filled job and follows the logic described in 2.7.

A decrease in I which corresponds to an increase in the share of legal immigrants λ would increase the expected labor cost and reduce the value of a vacancy to an unskilled-sector firm. Hence firms post fewer vacancies, the tightness of the labor market decreases putting downward pressure on wages and upward pressure on unemployment of native unskilled. 15

For both effects it is important to know whether the policy reducing I also reduces total immigrants (and their share in the labor force 1 − φ). A policy that decreases total immigrants (I + L) together with I may exacerbate both effects, while a policy that decreases I but increases (I + L) may attenuate and even reverse each effect. Before considering the general case it is useful to consider two special cases in which the price and the labor-cost effects work one at a time, while the other effect is muted. 3.1 Identical Options for Natives and Immigrants: the Price Channel only The first case considered is one in which unskilled natives, legal and illegal immigrants are identical in their search cost and in their probability of breaking up a match.

The parameter restrictions generating this case are: dI = dL = 0 (no probability of random return for immigrants) and πI = πL = 0 (no search costs for immigrants). In this case a decrease in I can be achieved through either border control or legalization (as the other two channels have been muted) and it will essentially represent a decrease in the supply of unskilled workers who are identical to native ones. While framed in a search-model with two labor markets (skilled and unskilled) the working of this model is very similar to that of a canonical model in which changing the number of illegal immigrants is like changing the supply of unskilled workers.

The effects on wages and employment are very similar to what a classical model of labor demand and supply for two complementary types of labor, would deliver.

A consequence of the assumptions above is that legal immigrants, illegal immigrants and native unskilled will be paid the same wage: wu 1N = wu 1L = wu 1I = wt 1. Therefore, the expected value of filling an unskilled vacancy with natives, legal or illegal immigrants is the same (JF,u 1N = JF 1L = JF 1I) and changing the share of legal, illegal immigrants and natives in the labor force has no effect on the incentive to post vacancies (the right-hand side of 17 does not depend on λ and φ in this case). This means that the labor-cost channel is not operating and the only effects work through the price channel.

3.1.1 Identical Natives and Immigrants: Effects of Border Controls A decrease in the number of illegal immigrants I achieved through increased border control (lower µI) reduces the total number of unskilled workers, (1 I) in country 1 and through the linear technology of the unskilled-sector it lowers Y u 1 = 16

m(θu 1 ) σu 1 +τ+m(θu 1 ) [1 L] in equilibrium. Since skilled and unskilled labor inputs are complements in the production of the final good (ρ < 1), the decrease in Y u 1 raises the marginal productivity of unskilled labor pu 1 and lowers that of skilled labor ps 1 (from 2 and 3). Since higher prices lead to higher surplus of a match, this induces the posting of unskilled jobs and raises the tightness and matching probability in the unskilled sector m(θu 1 ). The increase in the matching probability of unskilled native workers, in turn, drives their unemployment rate down and drives their wages up by improving their outside option.

The opposite holds for the skilled workers. Their unemployment rate increases and their wage decreases.

3.1.2 Identical Natives and Immigrants: Effects of Legalization: A decrease in the number of illegal immigrants I achieved through legalization (increase in the rate n) leaves the total number of immigrants unchanged by simply increasing the number of legal immigrants L by the same amount that it decreases illegal ones I. In this case “legal”and “illegal” are simply labels given to identical type of workers and they are also identical to unskilled natives. Hence legalization does not change any feature of the labor market nor the incentives of people in country 2 to immigrate since there is no benefit from obtaining legal status.

Hence in this case the production and the price of the unskilled intermediate input (Y u 1 and pu 1, respectively) remain unchanged. In this case, the legalization of illegal immigrants has no impact on job creation and labor market outcomes of native. Relative to the restrictive policy of increasing border controls, legalization fully eliminates the positive effects on wage and employment of unskilled natives and the negative effects on wage and employment of skilled native workers. 3.2 Perfect Substitution Skilled-Unskilled: the Labor-Cost channel only The second special case represents, in some respects, the opposite scenario.

In this case we consider perfect substitutability in production between skilled and unskilled workers (which corresponds to the assumption ρ = 1 in the production function 1) but we maintain differences between unskilled natives, legal immigrants and illegal immigrants so that dI > dL > 0 and πI > πL > 0. Illegal immigrants can be deported and they have the highest search costs. Legal immigrants have a certain probability of returning and also intermediate search costs.

In this case, the price effect is muted because the prices (marginal productivity) of the intermediate goods are constant, as the aggregate production function is linear in the intermediates. In particular ps 1 = α and pu and they will be unaffected by the relative supply of skilled and unskilled. This implies that the skilled sector is unaffected by the employment and labor market conditions in the unskilled sector, and as a consequence, the wage and unemployment rate of skilled native workers are independent of I. For unskilled workers, instead, the labor market effects of reducing illegal immigrants works only through their effects on the expected labor cost.

We can see from expression (17) that an increase in the proportion of natives in total unemployment of country 1 (φ) and an increase in the proportion of legal immigrants in the total number of unemployed immigrants (λ) decreases the expected value of a vacancy and reduces job creation7 .

Moreover, policies that increase the search cost for illegal immigrants (πI) or increase their deportation probability (dI) also influence directly the value of filling a vacancy with an illegal immigrant JF 1I and in turn affect the right-hand side of (17). Policies aimed at reducing illegal immigrants, therefore, can affect the expected labor cost to an unskilled firm in country 1 and in turn their job creation. 3.2.1 Perfect Skill Substitution: Border controls, Search Cost and Deportation Rates Border controls, search cost and deportation rates reduce the total proportion of immigrants in the unemployment pool of country 1 hence increasing φ.

This effect decreases the weight on term [λJF 1L + (1 − λ)JF 1I] and increases the weight on the term JF 1N in the right-hand side of (17). If JF 1I > JF 1L > JF 1N (which is the empirically relevant case) then the decline in the proportion of immigrants will increase the expected labor cost to an unskilled firm and decrease job creation and market tightness θu 1 to maintain the equality in (17).

Also, immigration policies aimed at reducing illegal immigrants would, increase λ, the fraction of legal workers among unemployed immigrants. Such a change shifts weight from JF 1I to JF 1L in the expression [λJF 1L + (1 − λ)JF 1I] of (17) and as long as JF 1I > JF 1L it reduces market tightness θu 1 to maintain equality. Both effects of the restrictive policies conjure to a decrease in θu 1 and hence they have an unambiguously positive impact on unemployment and negative impact on wages of unskilled native workers. 7 As long as expressions (15) and (16) are negative. 18

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